# Stochastic Optimal Control - Maximizing convex terminal costs

The theory of stochastic optimal control deals with the following problem:

Find $\quad\sup\limits_{u} \; \mathrm E[g(X^{(u,x)}_T)]$

where $X^{(u,x)}_t$ solves the following controlled SDE:

$dX_t=\sigma(u,X_t,t)dW$

$X_0 = x$ and $u(\omega,t)$ is the (adapted) control.

In my case:

$u(\omega,t)\in U$ for some convex set $U\subset\mathbb R$

and $g$ is a convex function!

My problem:

Most of the literature about stochastic optimal control, esp. maximum principles, deals with concave functions $g$ so that most of it does not apply to the convex case.

Is it, because the convex case the easier one and I just do not see how/why?

Where can I find something about the convex case?

Thanks,

Johannes

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I think that if your question is "it is interesting to know a function is convex to maximise it" you should not add complications related to stochastic process, optimal control... this will maximise the chance to have an interesting answer :) – robin girard Nov 23 '10 at 12:27
well the problem is a stochastic control problem, and I cannot leave anything out. I already know, that the supremum of a convex function is infinity, but in the problem presentet here, one is looking for the optimal control $u^*$ which maximises an expection of a stochastic process. – Johannes Nov 23 '10 at 16:16
@Johannes The supremum of a convex function is not necessarily infinity $sup_{x\in [0,1]}x^2=1$ ;). – robin girard Nov 24 '10 at 7:25